/.

Frank W. King

2009

The planar /. body problem (pronounced "slash dot") is the gravitational interaction of a line mass (/) and a point mass (.). The force and torque on the
line integrate exactly, facilitating analysis. The elongated asteroid Ida and its
tiny moonlet Dactyl form a natural example.

To study the /. body problem we used an advanced form of numeric integration called Symplectic Integration to take advantage of the symmetry of the
positions and velocities in our equations. This also helped manage the numerical
instability that we found in our equations. We prove that such an instability is
not an essential feature of the equations of the system but a numerical artifact.
We combine this sophisticated programming approach with cluster computing
to collect large amounts of data on this system that exhibits a rich array of
behavior.

The /. body problem realizes the complexity of the 3-body problem with only
2 bodies. In parameter space, sequences of periodic orbits dot a background of
chaotic orbits. We find known behaviors such a stabilization by gyroscopic motion and gravitationally stable orbits. Typically, the point and the line revolve
(= orbit) in precessing ellipses, as expected for a perturbation of the classical
2-body problem. However, the line may also rotate (= spin) chaotically or periodically. Spin-orbit momentum transfer orbits can spin-up the line or unbind
the point from the line, with applications in the statistics of asteroid rotation
rates.

Though our computation grid has collected a large amount of data which
containing the various periodic and chaotic behaviors, it has searched but a
small portion of the available parameter space. Our research has given us a
glimpse into the complex behavior of this system, and, from this, we can see
that the system is ripe for further investigation.